Regular Iteration

Regular Iteration or regular iterate refer to the fractional iterate a holomorphic function that is holomorphic in vicinity it its fixed point . In general, a holomorphic function may have several fixed points, and the fractional iterates, regular at different fixed points, have no need to coincide .

Transfer equation

Regular Iteration may refer to the solution $F$ of the transfer equation

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z\!+\!1)=T(F(z))$

where $~~T$ is known holomorphic function, that has physical sense of Transfer function. Then the regular iteration indicates an algorithm that allows to construct the superfunction $~F~$, that asymptotically approaches $~L~$ in some directions (usually at $+\infty$ or at $-\infty$), in the form

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle F(z) = L+\sum_{n=1}^{N} a_n \varepsilon^n + o(\varepsilon^N)$ , where $\varepsilon=\exp(kz)$

for some constant $~k~$ and some constant coefficients $~a~$.

In TORI, the representation of a fractional iterate of any function $~T~$ through its Superfunction and the Abel function is considered as principal, as it allows to deal with functions that have no real fixed points (and, perhaps, no fixed points at all). However, for the transfer function $~T~$ with real fixed point, the fractional iterate, that is regular at this fixed point, can be defined, constructed and evaluated also through the Schroeder function (solution of the Schroeder equation ) and its inverse function. The reason use of superfuncitons and not the Schroeder function in the consideration of the regular iterate is mainly historical, "it happened so". The application in laser science of the Transfer equation is more transparent than that of the Schroeder equation. However, while a function $~T~$ has real fixed point $~L~$ and is holomorphic in vicinity of $~L~$, then the same can be done with the Schroeder function too, and the corresponding deduction is supposed to be loaded in future. While, the deduction through the Transfer equation is presented below.

Deduction

Let $T(L)=L$. Let $T\,'(L)\ne 1$. Then the solution of equation (1) is searched in form of the expansion (2) for some constant coefficients $a$ ; usually, especially for real-holomorphic Transfer function $T$, it is assumed that $a_1= \pm 1$.

Such a representation allows to express

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~\displaystyle F(z\!+\!1) = L+\sum_{n=1}^{N} a_n K^n \varepsilon^n + o(\varepsilon^N)$ ,

where $K=\exp(k)$ and $N$ is natural number.

The substitution of (2) and (3) into (1) give equation

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~\displaystyle T\!\left(L+\sum_{n=1}^N a_n \varepsilon^n + o(\varepsilon^N)\right)-

L-\sum_{n=1}^{N} a_n K^n \varepsilon^n + o(\varepsilon^N) = 0$

Expansion of the left hand side of equation (4) at $\varepsilon \rightarrow 0$ gives

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ K=T\,'(L)$

Then, $k=\log(K)=\log\big(T\,'(L)\big)$ may be the convenient choise.

Often, value $a_1=\pm 1$ gives the superfunction with appropriate asymptotical properties. This allows simple expression for other $a$. In particular, the coefficients with $\varepsilon^2$ and $\varepsilon^3$ in the left hand side of (4) should be equal to zero; this gives

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (6) ~ ~ ~ \mathrm{e}^{2k} a_2=T\,' a_2+{T\,}/{2}$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (7) ~ ~ ~ \mathrm{e}^{3k} a_3=T\,' a_3 + T\, a_2+{T\,'}/{6}$

and so on.

In the simple case, the equations for other $a$ have single solutions; this allow to extend the expansion of the superfunciotn to arbitrary large $N$. Usually, the series diverges; but at $\Re(k)\ne 0$, for fixed $N$, the representation

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (8) ~ ~ ~ F(z)=T\,^m(F(z\!-\!m))$

for some appropriate integer $m$ allows to get the required precision of the evaluation due the exponential decay of the resulting $\varepsilon$ at the replacement $z\mapsto z+m$. However, at $\Re(k)<0$, $m$ should be negative, and the inverse of the transfer function, id set, $T^{-1}$, should be implemented; for non-trivial $T$, the choice of the cut lines of $T^{-1}$ determines the behavior (and singularities) of the resulting superfunction.

For application of the regular iteration to tetration tet$_b$ at $1\!<\!b\!<\!\exp(1/\mathrm e)$, the natural choose of the cut of the logarithm leads to the set of parallel cut lines to the direction of imaginary axis, separated with distance

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9) ~ ~ ~ \displaystyle \left| \frac{2 \pi} {k}\right| = \left| \frac{2\pi} {L ~ \ln(b)} \right| $.

The Regular Iteration allows to express the tetration and other superfunctions of $\exp_b$ at $1\!<\!b\!<\!\exp(1/\mathrm e)$ , the superfunction of factorial , that of the logistic sequence and the holomorphic extension of the Collatz subsequence; as well as the superfunction of the Doya function, that can be interpreted as transfer function of an idealized saturable optical amplifier .

The scaling of $a_1$ woiuld caused the corresponding modification of other coefficients; this is equivalent of the displacement of the argument of the resulting superfunction with some constant. For superfunctions of $\exp_b$, id est, superexponentials, such a constant is choosen in such a way, the superrunction $\mathrm {tet}_b(0)=1$ in the case of tetration and the superfunction is equal to the minimal integer values still bigger than $L$, for the growing along the real axis superexponential. (However, the last choice may be not good for the analysis of the dependence of the resulting growing superexponential on value of the base $b$ in the interval $1\!<\!b\!<\!\exp(1/\mathrm e)$ ).

The asymptotic expansion can be obtained also for the Abel function $~G=F^{-1}~$, either inverting series for $~F~$, or making the asymptotical expansion for the Abel equation, considering $~G(L\!+\!\varepsilon)~$, where $~L~$ is fixed point of the Transfer funciton, and $~\varepsilon~$ is small parameter. Together, $~F~$ and $~G~$ determine the regular iterations of the transfer function $~T~$ at fixed point $~L~$.

Zooming and Schroeder

The regular iterate of a transfer function $~T~$ at its fixed point $~L~$ can be constructed also with the zooming function $~f~$ and Schroeder function $~g=f^{-1}~$ for the new transfer function $~t~$ such that

$~ t(z)=T(L\!+\!z)-L~$

solving the corresponding Zooming equation

$~ T\big(f(z)\big)= f(K z)~$

and the Schroeder equation

$~ g\big(T(z))= K g(z)~$

Then, the regular iterate appears as

$~ T^r(z)=L+t^r(z\!-\!L)=L+f\Big(r g\big(\cdot(z\!-\!L)\big)\Big) ~$

which is almost equivalent of the representation through the Superfunction $~F~$ and the Abel function $~G~$.

Properties

The superfunctions constructed with regular iteration, have the specific common feature, the exponential behavior is some direction. For real-holomorphic transfer function $T$, at real coefficient $a_1$, the superfunction decays to constant (fixed point of the transfer function) either in the right hand side of the complex plane, or in the left hand side. In the opposite direction, the superfunction may show complicated behavior, but may also asymptotically approach another fixed point, as the tet$_b$ at $1\!<\! b\!<\!\exp(1/ \mathrm e) ~ $ does.

The superfunction, constructed with the regular iteration, is periodic; the period

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (10) ~ ~ ~ \displaystyle \tau=\frac{2 \pi\mathrm i}{k} = \frac{2 \pi \mathrm i}{ \ln\left(f'(L)\right)}$

For real-holomorphic transfer function, the the complex map of such superfunction reproduces itself at the translations for $2\pi/k$ in the imaginary direction. In particular, a$1\!<b\!<\!\exp(1/ \mathrm e)$, the tetration $\mathrm{tet}_b$ is periodic with period

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (11) ~ ~ ~ \displaystyle \tau= \frac{2 \pi \mathrm i}{ \ln\left( L\, \ln(b) \right)}$

Such periodicity (and in particular, the case $b\!=\!\sqrt{2}$, whlle $L\!=\!2$ for the tetrational and $L\!=\!4$ for the growing superexponential) is considered in .

Similar periodicity may have the Abel function $~G=F^{-1}~$. However, the Abel functions, corresponding at different [fixed points]] have no need to have the same periods. In general, the SuperFunctions constructed with regular iterations at different fixed points are not equivalent , and, in general, different SuperFunctions have different periods. However, they may look similar at the plots versus real argument. In particular the two of SuperExponentials to base $\sqrt{2}$ considered in , the maximal deviation is of order of $10^{-24}$ along the whole real axis. Such a deep similarity bring to confusion those researchers who make conclusions only on the base of the numerical simulations along the real axis. However, the deviation is clearly seen for complex values of the argument. Similar resemblance along the real axis may tale place also for the iterates of the transfer function $~T~$, constructed at the different fixed points $~L~$.

Examples

Regular iteration is used to construct the super-exponentials (including tetration) to base $b<\exp(1/\mathrm e)=\exp^2(-1)$, and in particilar, for $b=\sqrt{2}~$ , the superFactorial (id est, superfunction of factorial) and the holomorphic extension of the logistic sequence , which is superfunction of the quadratic function $z\mapsto rz(1\!-\!z)$ for positive values of $r$.

WIth regular iteration, the Holomorphic extension of the Collatz subsequence can be evaluated.

Below one more example is considered with the regular iteration of the transfer function of the idealized laser amplifier.

Regular iteration of the Doya function

$T=\mathrm{Doya}$ and its approximations

In this section, the Doya function is treated as the transfer function.

Consider the transfer function $T(z)=\mathrm{Doya}(z)$. The Doya function expresses the normalized output of the simple laser amplifier versus the normalized input, assuming, that the coefficient of amplification of a weak signal is $\mathrm e\!=\!\exp(1)\!\approx\! 2.7$

In vicinity of the real axis (While $|\Im(z)| \!<\! \pi$), this function can be expressed through the LambertW,

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (12) ~ ~ ~ \displaystyle T(z)=\mathrm{LambertW}\Big( z~ \mathrm{e}^{z+1} \Big)$

For real positive argument, this function is shown in figure at right with thick green line. At large value of the argument, the function shows almost linear growth, $T(z)\approx z+1+O(1/z)$. However, for the Regular iteration, the expansion at zero is important;

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (13) ~ ~ ~ \displaystyle

T(z)=\mathrm{Doya}(z) = \,$ $\displaystyle \mathrm e z-\mathrm e \left(\mathrm e\!-\!1\right) z^2 +\frac{1}{2} \mathrm e \left(-4 \mathrm e\!+\!3 \mathrm e^{2}\!+\!1\right) z^3 -\frac{1}{6} \mathrm e \left(12 \mathrm e\!-\!27 \mathrm e^{2}\!+\!16 \mathrm e^{3}\!-\!1\right) z^4 $ $ +O\left(z^5\right)$

Truncation of the series gives the linear and the quadratic approximations of the transfer function. These approximations are shown in the figure at right with thin lines. The expansion determines the derivatives of the transfer function, determining parameters $k$, $K$ and coefficients $a$ in the expansion of superfunction:

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (14) ~ ~ ~ \displaystyle k\!=\!\ln(T'(0))\!=\!1 ~ ~$, $~ ~\varepsilon=e^z$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (15) ~ ~ ~ \displaystyle T(0)\!=\!-2 \mathrm e\,(\mathrm e\!-\!1)~$, $~a_2\!=\!-1$

primary approximations by (18) and (19), brown and red curves, and the iterations of $T$ by (17) for $n=1,2,3,4$. The exact superfunction $F$ by (20) is shown with black curve

Then, the expansion (3) gives the primary approximation of the superfunction

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (16) ~ ~ ~ \displaystyle \tilde F(x)\approx \exp(x) - \exp(2x) + ..$

for large negative values of $\Re(z)$. The primary approximation with single term in the expansion above is plotted with brown line in the figure at right; the primary approximation with two terms is plotted with red line.

Let $\tilde F$ be one of the primary approximation. Then, the precise approximation can be constructed with

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (17) ~ ~ ~ \displaystyle F(x)\approx T^n(\tilde F(x\!-\!n))$

For the two primary approximations,

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (18) ~ ~ ~ \displaystyle \tilde F(x)=\exp(x) ~$

and

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (19) ~ ~ ~ \displaystyle \tilde F(x)=\exp(x) - \exp(2x)$,

the four regular iterations are shown in figure at right. The approximation becomes precise at $n>x\!+\!2$.

For this example, the superfunction $F$ can be espressed through the Tania function (and also through the LambertW function):

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (20) ~ ~ ~ \displaystyle F(x)= \mathrm{Tania}(x\!-\!1)= \mathrm{LambertW}(\exp(x))$

This "exact" superfunction is plotted with the black curve. It gives the exact representation for the iterates of the Doya function,

$~ \mathrm{Doya}^r(z)=\mathrm{Tania}\Big(t+\mathrm{ArcTania}(z)\Big)$

In the similar way, the regular iteration can be used to approximate other superfunctions, in praticular, those, which are not yet expressed through the special function. (However, after to reveal the properties and to provide the efficient algorithm for the expansion, any superfunction can be declared as known function, or "special function", and correspondently treated in future analysis and applications.)

Generalization

The Regular iteration is not universal way to express the superfunction.

Terms with different $k$ (various solutions of equation $~\exp(k)\!=\!K~$) can be added to the right hand side of equation (2), giving the more complicated expansions and more complicated superfunction. It is expected that namely the regular iteration corresponds to the physicaly–meaningful solution of the transfer equation of a ralistic amplifier; at least, it is so for the exampe above with Doya transfer functions.

Case without real fixed points

Some transfer functions have no fixed points; $~f(z)=\exp(z)\!+\!z~$ is an example of such a function.

Even the fixed point exist, it may be not real; also, even if it is real, the derivative at this fixed point may happen to be unity, the example of such transfer function is the exponential to base $b\!=\!\exp(1/ \mathrm e)$; in this case the expansion (2) should be modified . In these cases, the regular iteration requires some generalization, or other methods (for example, the Iterated Cauchi) should be used for the construction of the superfunction .

However, the regular iteration is expected to work for the most of amplifiers, that convert zero to zero, having some range in vicinity of zero, where the output can be considered as proportional to the input.

Uniqueness

The solution of the Transfer equation (as well as the solution of the Abel equation) is not unique. If $F$ is solution of the Transfer equation (0), then another solution, id est, superfunction $\tilde F$ can be expressed as

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (21) ~ ~ ~ \displaystyle F(z)=F(z+\eta(z))$

where $\eta$ is periodic holomorphic function with period unity. For the uniqueness, the additional requirements on the range of holomorphism of the superfunction should be postulated.

There is conjecture that for a physical unidimensional distributed homegeneous systems (which in general can be called amplifier), the namely Regular Iteration, if it can be applied, gives the meaningful ("physical") solution .

Conclusion

Not every superfunction can be represented through the regular iteration above, but in many cases, the regular iteration provides the simple and efficient way of the evaluation.

The simplified version of this article exists ; the slideshow is also available .